Abstract
Combinatorial designs have long been used to design efficient statistical experiments. More recently, connections to the theory of cryptographic communications have emerged. Combinatorial designs have provided solutions to problems coming from signal processing, radar, error-correcting codes, optical orthogonal codes, and image processing. Further, the most elegant solutions have come from designs with prescribed automorphisms. In this paper, we focus on partial geometric designs, a generalization of the classical 2-design. These designs have recently been shown to produce two directed strongly regular graphs. We generalize the well-known Kramer–Mesner Theorem for 2-designs to partial geometric designs. We also construct infinite families of partial geometric designs admitting a group of automorphisms which acts regularly on the point set and semi-regularly on the block set. The designs are obtained from new constructions of partial geometric difference families. These families were recently introduced a generalization of both the classical difference family and the partial geometric difference set.
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Notes
A directed strongly regular graph with parameters \((\mathbf{{v, k, t}},\lambda , \mu )\) is, by definition, a directed graph on \(\mathbf{{v}}\) vertices without loops such that (i) every vertex has in-degree and out-degree \(\mathbf{{k}}\), (ii) every vertex x has \(\mathbf{t}\) out-neighbors that are also in-neighbors of x, and (iii) the number of directed paths of length 2 from a vertex x to another vertex y is \(\lambda \) if there is an edge from x to y, and is \(\mu \) if there is no edge from x to y [9].
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Communicated by K. T. Arasu.
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Nowak, K., Olmez, O. Partial geometric designs with prescribed automorphisms. Des. Codes Cryptogr. 80, 435–451 (2016). https://doi.org/10.1007/s10623-015-0111-5
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DOI: https://doi.org/10.1007/s10623-015-0111-5